Page 163 - Lindens Handbook of Batteries

6.16 PRINCIPLES OF OPERATION

ˆ
[See Eq. (6.30) for definition of N]. The volume averaged reaction rate R is calculated using a simi-
lar expression. The current densities for the charge transfer reactions in the Butler-Volmer expression
are all expressed per unit volume
nFη
sj
,
c,jjj
,
j = di 2 = ia exp  α  aj j s j ,   - exp   -α nFη   (6.40)
j 
 
dx 0,   RT   RT  
where a is the area available for reaction, per unit volume of the electrode. The subscript 2 for the
current density refers to the solution phase (i.e., the electrolyte).
6.7 LEAD-ACID BATTERY MODEL
16
A lead-acid battery model developed by Nguyen has the following reactions taking place:
r
PbO + 2 HSO + 4 - 3 H + + e 2 - → PbSO + 2 HO (positiveelectrode) (6.41)
2
4
+
Pb HSO → 4 - PbSO + 4 e + 2 - H + (negativeelectrode (6.42)
)
At each electrode, the material balance is given by Eq. (6.38). The reaction term R is given by the
k
Butler-Volmer equation for reaction [Eq. (6.41)] at the positive electrode and [Eq. (6.42)] at the nega-
tive electrode. In addition, the volume of the product formed in these reactions (i.e., PbSO ) is much
4
higher than the reactants (PbO or Pb). This induces a change in the porosity of the electrode during
2
charge/discharge. This change in porosity is accounted for by using Faraday’s law as follows: 17
∂ε 1   M   M   di j 

=     -      (6.43)
∂t n F    ρ  Product  ρ  Reactant    dx 
j
The total current is carried across the electrode, from the separator/electrode interface to the current-
collector/electrode interface, by both the electrons within the electrode matrix and by the ions in the
electrolyte that fill the pores across the thickness of the electrode (see Sections 6.4.1 and 6.4.2)
i
i tot =+ i (6.44)
1
2
The current across the electrode matrix (i ) is given by Eq. (6.12) and the current across the electro-
1
lyte (i ) is given by Eq. (6.18), after modifying the conductivity as shown in Eq. (6.37). In addition,
2
the influence of concentration gradients on the transport of ions is modeled using transport numbers.
The resultant expression for the current in the electrolyte phase is given by 16
- (
 2 RT 0 ∇ c 
i =- κ eff  ∇ 2 +φ 12 t )  (6.45)
2
+
 F c 
The resultant profiles for the distribution of the porosity across the thickness of the electrode are
shown in Fig. 6.9. The porosity at the electrode/separator interface decreases for both the anode
and the cathode, owing to the precipitation of PbSO that occupies a higher volume than the active
4
materials. The reduction in porosity is higher at the negative electrode since the difference in density
between the reactant (metallic lead) and the product (PbSO ) is higher. A direct effect of blocking
4
the electrode surface is the restriction of access of the entire volume of the electrode to the electro-
lyte. As a result, the reaction distribution is highly nonuniform, as shown in Fig.6.10. The reaction
rates at the current-collector end of the electrodes are close to zero, indicating poor utilization of
the electrodes.